Differential Equations

Equations which are composed of an unknown function and its derivatives.

Types

Ordinary Differential Equations

When a differential equation involves one independent variable, and one or more dependent variables.

An example:

$$\frac{\text{d}y}{\text{d}x} = \cos(x)$$

Partial Differential Equations

When a differential equation involves more than one independent variables, and more than one dependent variables.

$$\frac{\partial y}{\partial x} = y_x = \cos(x)$$

Linear

A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function (dependant variable) and its derivatives, that is an equation of the form:

$$P_0 (x) y + P_1 (x) y' +\,...\,+ P_n (x) y^{(n)} + Q(x) = 0$$

Here:

• $P_0, P_1,\dots,P_n,Q$ are all differentiable functions of $x$, doesn’t depend on $y$
• $y(x)$ is the unknown function
• $y^{(n)}$ denotes the $n$th derivative of $y$

Nonlinear

Nonlinear differential equations are any equations that cannot be written in the above form. In particular, these include all equations that include:

• $y$ and/or its derivatives raised to any power other than $1$
• nonlinear functions of $y$ or any of its derivative
• any product or function of these

Properties of Differential Equations

Order

Highest order derivative.

Degree

Power of highest order derivative.

Initial Value Problem (IVP)

A differential equation along with appropriate number of initial conditions.

Initial condition(s) is/are required to determine which solution (out of the infinite number of solutions) is the suitable one for the given problem.

Picard’s Existence and Uniqueness Theorem

Consider the below IVP.

$$\frac{\text{d}y}{\text{d}x} = f(x,y) \;;\; y(x_0)=y_0$$

Suppose: $D$ is an open neighbourhood in $\mathbb{R}^2$ containing the point $(x_0,y_0)$.

If $f$ and $\frac{\partial{f}}{\partial{y}}$ are continuous functions in $D$, then the IVP has a unique solution in some closed interval containing $x_0$.